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1.
The unsteady flow of viscoelastic fluid with the fractional derivative Maxwell model (FDMM) in a channel is studied in this note. The exact solutions are obtained for an arbitrary pressure gradient by means of the finite Fourier cosine transform and the Laplace transform. Two special cases of pressure gradient are discussed. Some results given by the classical models with integer-order are included in this note. 相似文献
2.
The unsteady flow of an incompressible fractional Maxwell fluid between two infinite coaxial cylinders is studied by means of integral transforms.The motion of the fluid is due to the inner cylinder that applies a time dependent torsional shear to the fluid.The exact solutions for velocity and shear stress are presented in series form in terms of some generalized functions.They can easily be particularized to give similar solutions for Maxwell and Newtonian fluids.Finally,the influence of pertinent parameters on the fluid motion,as well as a comparison between models,is highlighted by graphical illustrations. 相似文献
3.
The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied. 相似文献
4.
Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model 总被引:6,自引:0,他引:6
The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. The flow near
a wall suddenly set in motion is studied for a non-Newtonian viscoelastic fluid with the fractional Maxwell model. Exact solutions
of velocity and stress are obtained by using the discrete inverse Laplace transform of the sequential fractional derivatives.
It is found that the effect of the fractional orders in the constitutive relationship on the flow field is significant. The
results show that for small times there are appreciable viscoelastic effects on the shear stress at the plate, for large times
the viscoelastic effects become weak.
The project supported by the National Natural Science Foundation of China (10002003), Foundation for University Key Teacher
by the Ministry of Education, Research Fund for the Doctoral Program of Higher Education 相似文献
5.
Attila Pálfalvi 《International Journal of Non》2010,45(2):169-175
Fractional order (or, shortly, fractional) derivatives are used in viscoelasticity since the late 1980s, and they grow more and more popular nowadays. However, their efficient numerical calculation is non-trivial, because, unlike integer-order derivatives, they require evaluation of history integrals in every time step. Several authors tried to overcome this difficulty, either by simplifying these integrals or by avoiding them. In this paper, the Adomian decomposition method is applied on a fractionally damped mechanical oscillator for a sine excitation, and the analytical solution of the problem is found. Also, a series expansion is derived which proves very efficient for calculations of transients of fractional vibration systems. Numerical examples are included. 相似文献
6.
In this note the velocity field and the adequate shear stress corresponding to the unsteady flow of a fractional Maxwell fluid due to a constantly accelerating circular cylinder have been determined by means of the Laplace and finite Hankel transforms.The obtained solutions satisfy all imposed initial and boundary conditions.They can easily be reduced to give similar solutions for ordinary Maxwell and Newtonian fluids.Finally,the influence of pertinent parameters on the fluid motion,as well as a comparison between models,is underlined by graphical illustrations. 相似文献
7.
Correcting the initialization of models with fractional derivatives via history-dependent conditions 总被引:1,自引:0,他引:1
Fractional differential equations are more and more used in modeling memory(history-dependent,nonlocal,or hereditary) phenomena.Conventional initial values of fractional differential equations are define at a point,while recent works defin initial conditions over histories.We prove that the conventional initialization of fractional differential equations with a Riemann–Liouville derivative is wrong with a simple counter-example.The initial values were assumed to be arbitrarily given for a typical fractional differential equation,but we fin one of these values can only be zero.We show that fractional differential equations are of infinit dimensions,and the initial conditions,initial histories,are define as functions over intervals.We obtain the equivalent integral equation for Caputo case.With a simple fractional model of materials,we illustrate that the recovery behavior is correct with the initial creep history,but is wrong with initial values at the starting point of the recovery.We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically. 相似文献
8.
Dr. Lucas Duffrène René Gy Hélène Burlet Roland Piques Annelise Faivre Anas Sekkat Joseph Perez 《Rheologica Acta》1997,36(2):173-186
The linear viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading is investigated in the glass transition range. Using the time-temperature superposition technique, the master curves of the shear dynamic relaxation moduli are obtained at a reference temperature of 566°C. A method to determine the viscoelastic constants from dynamic relaxation moduli is proposed. However, some viscoelastic constants cannot be directly measured from the experimental curves and others cannot be precisely obtained due to non-linearity effects at very low frequencies. The generalized Maxwell model is investigated from the experimental dynamic moduli without fixing the viscoelastic constants. A set of parameters is shown to be in good agreement with the experimental dynamic relaxation moduli, but does not give the correct values of the viscoelastic constants of the investigated glass. The soda-lime-silica glass exhibits a non-linear viscoelastic behavior at very low stress level which is usually observed for organic glasses. This non-linear behavior is questioned. 相似文献
9.
The fractional calculus is used in the constitutive relationship model of viscoelastic fluid. A generalized Maxwell model with fractional calculus is considered. Based on the flow conditions described, two flow cases are solved and the exact solutions are obtained by using the Weber transform and the Laplace transform for fractional calculus.The project supported by the National Natural Science Foundation of China (10272067, 10426024), the Doctoral Program Foundation of the Education Ministry of China (20030422046) and the Natural Science Foundation of Shandong University at Weihai. The English text was polished by Keren Wang. 相似文献
10.
Recently, [Rao, I.J., Rajagopal, K.R., 2007. Status of the K-BKZ model within the framework of materials with multiple natural configurations. Journal of Non-Newtonian Fluid Mechanics, 141, 79–84] showed that the K-BKZ Model is a special sub-class of models based on a thermodynamic framework that takes into account the fact that bodies are capable of existing stress free in multiple configurations with special choices being made for the way in which the body stores energy and the way it dissipates energy. They also showed that several generalizations of the K-BKZ model are possible. In this short note we show that two distinct methods of storing energy and dissipating energy lead to the classical Maxwell model. That is, in addition to the classical choice for the storage of energy and rate of dissipation (the usual spring dashpot analogy) a more complicated choice also leads to the same model. This result is rather important as it shows that a variety of means for storing and dissipating energy can lead to the same mechanical response, when one restricts oneself to purely mechanical considerations. 相似文献
11.
This paper introduces a new model for the Fourier law of heat conduction with the time-fractional order to the generalized Maxwell fluid. The flow is influenced by magnetic field, radiation heat, and heat source. A fractional calculus approach is used to establish the constitutive relationship coupling model of a viscoelastic fluid. We use the Laplace transform and solve ordinary differential equations with a matrix form to obtain the velocity and temperature in the Laplace domain. To obtain solutions from the Laplace space back to the original space, the numerical inversion of the Laplace transform is used. According to the results and graphs, a new theory can be constructed. Comparisons of the associated parameters and the corresponding flow and heat transfer characteristics are presented and analyzed in detail. 相似文献
12.
During the last two decades fractional calculus has been increasingly applied to physics, especially to rheology. It is well known that there are obivious differences between Riemann-Liouville (R-L) definition and Caputo definition, which are the two most commonly used definitions of fractional derivatives. The multiple definitions of fractional derivatives have hindered the application of fractional calculus in rheology. In this paper, we clarify that the R-L definition and Caputo definition are both rheologically unreasonable with the help of the mechanical analogues of the fractional element model. We also find that to make them more reasonable rheologically, the lower terminals of both definitions should be put to ?∞. We further prove that the R-L definition with lower terminal ?∞ and the Caputo definition with lower terminal ?∞ are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points. Thus we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal ?∞ (or, equivalently, the Caputo derivatives with lower terminal ?∞ ) not only for steady-state processes, but also for transient processes. 相似文献
13.
A fractional-order Maxwell model is used to describe the viscoelastic seabed mud.The experimental data of the real mud well fit the results of the fractional-order Maxwell model that has fewer parameters than the traditional model.The model is then used to investigate the effect of the mud on the surface-wave damping.The damping rate of a linear monochromatic wave is obtained.The elastic resonance of the mud layer is observed,which leads to the peaks in the damping rate.The damping rate is a sum of the m... 相似文献
14.
A generalization of the Maxwell model for polymer systems is derived that replaces the velocity gradient in the Eulerian expression for the upper convected derivative by a tensorial kinematic function. Applying the principle of objectivity this tensorial function is reduced to two scalar slip functions. In shear flows, only one of the two occurs. Material functions are calculated in closed form, and asymptotic conditions are formulated that guarantee isotropic behaviour of the material in sudden strains.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada. 相似文献
15.
R. Vilela Mendes 《Nonlinear dynamics》2009,55(4):395-399
Based on criteria of mathematical simplicity and consistency with empirical market data, a model with volatility driven by
fractional noise has been constructed which provides a fairly accurate mathematical parametrization of the data. Here, the
model is formulated in terms of a fractional integration of stochastic processes. 相似文献
16.
The theory of fractional calculus goes back to the beginning of the theory of differential calculus, but its application received
attention only recently. In the area of automatic control some work was developed, but the proposed algorithms are still in
a research stage. This paper discusses a novel method, with two degrees of freedom, for the design of fractional discrete-time
derivatives. The performance of several approximations of fractional derivatives is investigated in the perspective of nonlinear
system control. 相似文献
17.
Accurate prediction of coal׳s creep behavior is of great significance to coalbed methane extraction. In this study, taking into account the visco-elastic–plastic characteristics and the damage effect, a fractional non-linear model is proposed to describe the creep behavior of coal. The constitutive and creep equations of the proposed fractional non-linear model are derived via the Boltzmann superposition principle and discrete inverse Laplace transform. Furthermore, uniaxial creep tests under different axial stress conditions were carried out to validate the proposed model. It is found that the present model can describe the experimental data from creep tests with better accuracy than classical models. Particularly, the present model can predict the accelerating creep deformation of coal which classical models fail to reproduce. Finally, the parametric sensitivity analysis is performed to investigate the effects of model parameters on the creep strain. It is verified that the introduction of fractional parameters and damage factor in the present model is essential to accurate prediction of the full creep stage of coal. 相似文献
18.
Hong Zhou Wei Kang Arthur Krener Hongyun Wang 《ournal of non Newtonian Fluid Mechanics》2008,150(2-3):104-115
The controllability of viscoelastic fields is a fundamental concept that defines some essential capabilities and limitations of the resulting materials. In this paper, we study the controllability of different homogeneous flow fields of viscoelastic fluids governed by the upper convected Maxwell model. The approach is largely based on the nonlinear geometric control theory. Through the analysis of the control Lie algebra, we find the submanifolds in the state space on which the homogeneous flow fields are weakly controllable. Our approach can be generalized to more complicated systems. 相似文献
19.
In this paper, the first-passage failure of stochastic dynamical systems with fractional derivative and power-form restoring force subjected to Gaussian white-noise excitation is investigated. With application of the stochastic averaging method of quasi-Hamiltonian system, the system energy process will converge weakly to an Itô differential equation. After that, Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time are constructed and solved. Particularly, the influence on reliability caused by the order of fractional derivative and the power of restoring force are also examined, respectively. Numerical results show that reliability function is decreased with respect to the time. Lower power of restoring force may lead the system to more unstable evolution and lead first passage easy to happen. Besides, more viscous material described by fractional derivative may have higher reliability. Moreover, the analytical results are all in good agreement with Monte-Carlo data. 相似文献
20.
The stress response of amorphous polymers exhibits tremendous change during the glass transition region, from soft viscoelastic response to stiff viscoplastic response. In order to describe the temperature-dependent and rate-dependent stress response of amorphous polymers, we extend the one-dimensional small strain fractional Zener model to the three-dimensional finite deformation model. The Eyring model is adopted to represent the stress-activated viscous flow. A phenomenological evolution equation of yield strength is used to describe the strain softening behaviors. We demonstrate that the stress response predicted by the three-dimensional model is consistent with that of one-dimensional model under uniaxial deformation, which confirms the validity of the extension. The model is then applied to describe the stress response of an amorphous thermoset at various temperatures and strain rates, which shows good agreement between experiments and simulation. We further perform a parameter study to investigate the influence of the model parameters on the stress response. The results show that a smaller fractional order results in a larger yield strain while has little effect on the yield stress when the temperature is below the glass transition temperature. For the stress relaxation tests, a smaller fractional order leads to a slower relaxation rate. 相似文献